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We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs…
Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided…
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of…
A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the…
In this article, we present that for any complex manifold whose dimension is bigger than one, there exists a multiplier ideal sheaf such that there don't exist equisingular weights with logarithmic poles, which are not smaller than the…
Throughout our work on the L\^e cycles of an affine hypersurface singularity, our primary algebraic tool consisted of a method for taking the Jacobian ideal of a complex analytic function and decomposing it into pure-dimensional "pieces".…
Stochastic differentiable approximation schemes are widely used for solving high dimensional problems. Most of existing methods satisfy some desirable properties, including conditional descent inequalities, and almost sure (a.s.)…
Every non-solvable and non-semisimple quadratic Lie algebra can be obtained as a double extension of a solvable quadratic Lie algebra. Thanks to a partial classification of nilpotent Lie algebras and this result, we can design different…
We present a theorem of resolution of singularities for real analytic constrained differential systems $A(x)\dot{x} = F(x)$ defined on a 2-manifold with corners having impasse set $\{x; \det A(x) = 0\}$. This result can be seen as a…
We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods…
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of $m$ linear inequalities in $n$ variables $(P): A^{\top}x \le u$ when its set of solutions has positive volume. However, when $(P)$ is infeasible,…
Differential resultant formulas are defined, for a system $\mathcal{P}$ of $n$ ordinary Laurent differential polynomials in $n-1$ differential variables. These are determinants of coefficient matrices of an extended system of polynomials…
We explicitly compute the maximal Lyapunov exponent for a switched system on $\mathrm{SL}_2(\mathbb R)$. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with…
The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's…
Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial…
A reduced divisor on a nonsingular variety defines the sheaf of logarithmic 1-forms. We introduce a certain coherent sheaf whose double dual coincides with this sheaf. It has some nice properties, for example, the residue exact sequence…
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an…
An existing solvability result for relaxed one-sided Lipschitz algebraic inclusions is substantially improved. This enhanced solvability result allows the design of a very robust numerical method for the approximation of a solution of the…
Let $G\subseteq GL(n)$ be a finite group without pseudo-reflections. We present an algorithm to compute and verify a candidate for the Cox ring of a resolution $X\rightarrow \mathbb{C}^n/G$, which is based just on the geometry of the…